Quotients of Sums and Differences of Perfect Squares 13 



We expand the numerator of this quotient in terms of x by 

 repeated application of [i] and find 



P(ai, . . .,ak-i,x,ajs:+i, . . .,ak+ux,ak-i, . . .,«i) = Bx''--i-2Gx-\-C, 



where 



2 2 



B ^= Pi,k-\pk+\,k+\. G =^ pi,k~\pk+\,k+2-\-p\,k-\p\,k-lpk+hk-\-\, 



jL 2 2 



C = 2PiJt-^ipi^ji:-2Pk+l,k+2-{-Pl,k-lpk+2,k+2-\'Pl,k-2Pk+l,k+l' 



Similarly 



p(a2,. . .,ak-\,x,ak-\-\,. . .,ak+i,x,ak-i, . . .,a-2) = Ax^+2//x+2F, 



where 



2 2 



A = P2,k-lPk+l,t+l, -^ = P"lk-lpk+l,k+2-\-p2,k-lp2,k-2pk+l,k+U 



i 2 2 



^ = -2(.'2P2,k-lp2,k-2pk+l,k+2-'rP2,k~lPk+2,k+2-{'p2,k-2pk+l,k+l- 



It follows that the integral values of [25] for integral values of 

 X are given by the integral solutions of the equation 



<^ = ^4x-j'—Bx^-i-2//xy'-2Gx^2F}'—C ~ o [26] 



For all values of ^>i, this is a cubic form, the discussion of 

 which can not be brought within the scope of a short paper, 

 but for ^=1, that is when the unknowTi element occupies the 

 first place in the continiiant of the numerator, we have, by virtue 

 of [2] 



^ = A0A2 = A2. <^ =A.oA,3+A.oA,-iA2 = As. 



2 2 2 



^ = ^Pl,0pl,~iPs.S~^Ph0p3,3-\-pl.-lp2.2 = As- ^ = AcA.2= O. 



2 



-^^ ACA3+A0A-1A2 = o. 



2 ' 



2F= 2/>2,o/>2,-lA3+A.oA3+A-lA2 =A2. 



since 



Ai = A-o = I' Ao =P\-\ = o. A-i = I' 

 In this case we have therefore 



-<j> = p22^^-\-2p,,^x—p,,,y-\-p,,, = o, 

 367 



